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Weak approximation for stochastic reaction-diffusion equation near sharp interface limit

Jianbo Cui, Liying Sun

TL;DR

The paper develops a weak error analysis for stochastic reaction-diffusion equations near sharp interface limit by introducing a regularized problem that is exponentially ergodic and by deriving regularity for the regularized Kolmogorov and Poisson equations with polynomial dependence on $\frac{1}{\epsilon}$. A parameterized splitting scheme is analyzed via continuous interpolation, yielding weak error bounds that grow only polynomially in $\frac{1}{\epsilon}$ and, under further assumptions, an $\epsilon$-dependent rate that vanishes with suitably small time steps $\tau$. It also establishes convergence of invariant measures in total variation and proves a numerical central limit theorem for time-averaged observables under the splitting scheme. The approach leverages averaged noise-regularization effects, long-time ergodicity, and Poisson-regularity to extend to other SPDEs without relying on spectral estimates. Overall, the results offer a practical and rigorous pathway to simulate sharp-interface stochastic dynamics with controlled $\epsilon$-dependence.

Abstract

It is known that when the diffuse interface thickness $ε$ vanishes, the sharp interface limit of the stochastic reaction-diffusion equation is formally a stochastic geometric flow. To capture and simulate such geometric flow, it is crucial to develop numerical approximations whose error bounds depends on $\frac 1ε$ polynomially. However, due to loss of spectral estimate of the linearized stochastic reaction-diffusion equation, how to get such error bound of numerical approximation has been an open problem. In this paper, we solve this weak error bound problem for stochastic reaction-diffusion equations near sharp interface limit. We first introduce a regularized problem which enjoys the exponential ergodicity. Then we present the regularity analysis of the regularized Kolmogorov and Poisson equations which only depends on $\frac 1ε$ polynomially. Furthermore, we establish such weak error bound. This phenomenon could be viewed as a kind of the regularization effect of noise on the numerical approximation of stochastic partial differential equation (SPDE). As a by-product, a central limit theorem of the weak approximation is shown near sharp interface limit. Our method of proof could be extended to a number of other spatial and temporal numerical approximations for semilinear SPDEs.

Weak approximation for stochastic reaction-diffusion equation near sharp interface limit

TL;DR

The paper develops a weak error analysis for stochastic reaction-diffusion equations near sharp interface limit by introducing a regularized problem that is exponentially ergodic and by deriving regularity for the regularized Kolmogorov and Poisson equations with polynomial dependence on . A parameterized splitting scheme is analyzed via continuous interpolation, yielding weak error bounds that grow only polynomially in and, under further assumptions, an -dependent rate that vanishes with suitably small time steps . It also establishes convergence of invariant measures in total variation and proves a numerical central limit theorem for time-averaged observables under the splitting scheme. The approach leverages averaged noise-regularization effects, long-time ergodicity, and Poisson-regularity to extend to other SPDEs without relying on spectral estimates. Overall, the results offer a practical and rigorous pathway to simulate sharp-interface stochastic dynamics with controlled -dependence.

Abstract

It is known that when the diffuse interface thickness vanishes, the sharp interface limit of the stochastic reaction-diffusion equation is formally a stochastic geometric flow. To capture and simulate such geometric flow, it is crucial to develop numerical approximations whose error bounds depends on polynomially. However, due to loss of spectral estimate of the linearized stochastic reaction-diffusion equation, how to get such error bound of numerical approximation has been an open problem. In this paper, we solve this weak error bound problem for stochastic reaction-diffusion equations near sharp interface limit. We first introduce a regularized problem which enjoys the exponential ergodicity. Then we present the regularity analysis of the regularized Kolmogorov and Poisson equations which only depends on polynomially. Furthermore, we establish such weak error bound. This phenomenon could be viewed as a kind of the regularization effect of noise on the numerical approximation of stochastic partial differential equation (SPDE). As a by-product, a central limit theorem of the weak approximation is shown near sharp interface limit. Our method of proof could be extended to a number of other spatial and temporal numerical approximations for semilinear SPDEs.
Paper Structure (23 sections, 22 theorems, 233 equations)

This paper contains 23 sections, 22 theorems, 233 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Properties of the original problem
  4. Phase flow of parameterised ODE
  5. Properties of the regularized problem
  6. Temporal approximation
  7. Splitting numerical approximation
  8. A priori bound of splitting numerical approximation
  9. Kolmogorov equation
  10. Kolmogorov equation
  11. Longtime regularity estimate of Kolmogorov equation via ergodicity
  12. Application to the weak approximation
  13. Continuous interpolation of splitting scheme
  14. Polynomial dependence on $\frac{1}{\epsilon}$ in weak error analysis
  15. Convergence in total variation distance
  16. ...and 8 more sections

Key Result

Theorem 1.1

Let Assumption ap-1 and erg-noi hold and $\phi\in \mathcal{C}_b^2(\mathbb H).$ Let $0<c_1\le \epsilon\lambda \le c_2<\infty$ for postive constants $c_1$ and $c_2.$ Assume that $\tau(\lambda+\frac{1}{\epsilon})\sim O(1)$. Then the weak error of $\{u_k\}_{k\in \mathbb N}$ with $u_0=x$ satisfies the fo

Theorems & Definitions (49)

  • Theorem 1.1
  • Lemma 2.2
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Example 2.5
  • Lemma 2.6
  • Remark 2.7
  • Lemma 3.1
  • Lemma 3.2
  • ...and 39 more